FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 217 



We know that the vectors of the theory are connected with one another and the 

 actual co-ordinates of the system hy the equations 



div E = 47r2e-47r div P 



, 47T dP 47T 



= e ~dt + T d* + T 



dl 



In these equations P is the dielectric and I the magnetic polarisation intensity ; f m is 

 the velocity and r m the position vector of the' element of matter and r e and ? the 

 velocity and position vectors relative to this element of the typical element of 

 free charge (e) over which the sum 2 in the first and second equations is taken 

 per unit volume at each place. 



In these equations we have purposely refrained from assuming a deh'nite electronic 

 constitution for the dielectric and magnetic polarisations as it was desired to emphasise 

 certain points in connexion with the mechanical forcive which have not yet been 

 adequately dealt with. 



We have thus to introduce three undetermined multipliers one scalar </> and two 

 vectors A,, A 3 and it is then the variation of 



f ''dt \ 



J I, J 



iv P) 



O7T 47T 



dP 



G 



dl 



1 / A i TT I dEi 4?r dP 4ir ^ , m . -, 



-- A,, Curl H --- r -^ --- Curl Pr,.. 



47T\ G dt G dt C 



that is to he made null, afterwards determining the forms of the various undetermined 

 functions to satisfy the restrictions which necessitated their introduction. In 

 conducting the variation we can now treat the electric force, displacement and 

 polarisation, the magnetic force induction and polarisation and the position co-ordinates 

 of the electrical and material elements as all independent. We here see the reason 

 for introducing the equation expressing the rate of change of H instead of the 



equation 



H = B-47rI 



determining its value, for this latter equation does not in reality enable us to obtain 

 a relation between the variations of H and the position co-ordinates of the matter, so 

 we could not treat all our variables as independent. 



2 H 2 



